econ110_cheat_sheet_final_-_yale - Exercise A monopolist...

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Exercise A monopolist faces the demand curve Q = 120 - 3P . Find the revenue maximizing price and quantity: Maximizing revenue is equivalent to maximizing profit when MC = 0. By the (P-choke + MC)/2 rule, the optimal price is 20. Then, the optimal quantity is 60. Exercise Find the profits at the Nash equilibrium . Consider two firms that play a simultaneous move Cournot competition game . The market inverse demand is given by P = 120 – Q . The marginal cost is zero for both firms: P1 = 120 – Q1 – Q2 … MR1 = 120 – Q1 – 2(Q2) = 0 … solve for Q1, then set Q1 and Q2 = Qne, solve … obtain Qne and Pne, solve for profits. Profits = (Pne – MC) x Qne Exercise Consider now a Cournot game where each firm is allowed to choose either the Nash equilibrium quantity or half the monopoly quantity, only. Draw the normal form of the simultaneous move game. By the P-choke/MC rule the monopoly profit- maximizing price is 60, which makes the monopoly quantity 60. If each firm produces half of that, each makes a profit of 1800. If one firm produces half the monopoly quantity and the other produces the Nash equilibrium outcome, the price is 50: (P = 120 – Qnash - .5Qmonop) The profits are 1500 and 2000, respectively: Nash= (50-0)(40) .5 Monop= (50-0)(30) Nash Half Monopoly Nash 1600, 1600 1500, 2000 Half Monopoly 2000, 1500 1800, 1800 Exercise Argue why the game above is a prisoner’s dilemma: Each firm has a dominant strategy. Also, the dominant strategy equilibrium is pareto inferior to (1800, 1800). Exercise A monopolist produces at constant marginal cost and faces two segmented markets with the following demand curves: Qa = 100- P … Qb = 100 – 2(P) In which market will he charge a higher price if he maximizes profits? : E = P/(Pchoke - P) The first demand has a higher choke price and is therefore more inelastic for any price. By the mark up elasticity formula: the lower the elasticity, the higher the price you can charge, w/o the quantity changing that much. Therefore, the price should be higher in segment A. Exercise Assume that you estimate the elasticity of demand for luxury cars to be 0.8. On the basis of this information, can you conclude that at least one car firm is not maximizing profits? Explain: It is true that any firm that maximizes profits will produce at the elastic (E > 1) part of the demand. However, the estimate of 0.8 refers to the industry demand…isn’t informative about the pricing behavior of individual firms. Exercise Consider the Stackelberg ( sequential ) version of a two-player Cournot competition game. Would you rather be the follower, or the leader? (Assume that both firms have the same constant marginal cost.): We know that in the Stackelberg version of the Cournot game, the following statements hold, relative to the Nash equilibrium outcome of the simultaneous move game: The leader wants the follower to decrease his quantity, because the game has negative externalities. The leader achieves this by increasing his quantity, because actions are strategic substitutes. This lowers the profits of the follower, because the game has negative externalities. The profit of the leader goes up in all games. Therefore, the leader has higher profits than the follower and you
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